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Yukti Vault

Algebra

Matrices: JEE Main & Advanced PYQs

203 previous year questions on Matrices from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 203

  1. Let $A=\begin{pmatrix}4&-2\\\alpha&\beta\end{pmatrix}$. If $A^2+\gamma A+18I=O$, then $\det(A)$ is equal to
  2. Let the matrix $A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$ satisfy $A^{n}=A^{n-2}+A^{2}-I$ for $n\ge3$. Then the sum of all the elements of $A^{50}$ is :
  3. Let $A$ be a $3\times3$ real matrix such that $A\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\1\\0\end{pmatrix}$ …
  4. Let $A$ be a $3\times3$ real matrix such that $A^{2}(A-2I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If …
  5. Let $B=\begin{bmatrix}1&3\\1&5\end{bmatrix}$ and $A$ be a $2\times2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^{4}+\alpha C^{2}+\beta I=O$, then …
  6. Let $A=\begin{bmatrix}1&2\\-1&4\end{bmatrix}$. If $A^{-1}=\alpha I+\beta A$, $\alpha,\beta\in\mathbf{R}$, $I$ is a $2\times2$ identity matrix, then $4(\alpha-\beta)$ is …
  7. If $A=\begin{pmatrix}2&2\\9&4\end{pmatrix}$ and $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$, then $10A^{-1}$ is equal to :
  8. Let $A=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $B=\begin{bmatrix}9^2&-10^2&11^2\\12^2&13^2&-14^2\\-15^2&16^2&17^2\end{bmatrix}$, then the value of $A'BA$ is:
  9. Let $\theta=\frac\pi5$ and $A=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}$. If $B=A+A^4$, then $\det(B)$ :
  10. Let $A=\begin{bmatrix}x&3&2\\1&y&4\\2&2&z\end{bmatrix}$, $xyz=60$ and $8x+4y+3z=20$. If $A(\operatorname{adj}A)=kI$, then $k$ is equal to
  11. Let $S=\left\{\begin{pmatrix}-1&a\\0&b\end{pmatrix}:a,b\in\{1,2,3,\ldots,100\}\right\}$ and let $T_n=\{A\in S:A^{n(n+1)}=I\}$. Then the number of elements in …
  12. Let $A=\begin{bmatrix}1&a&a\\0&1&b\\0&0&1\end{bmatrix}$, $a,b\in\mathbb{R}$. If for some $n\in\mathbb{N}$, $A^n=\begin{bmatrix}1&48&2160\\0&1&96\\0&0&1\end{bmatrix}$ …
  13. Let $X=\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}$, $Y=\alpha I+\beta X+\gamma X^{2}$ and $Z=\alpha^{2}I-\alpha\beta X+\left(\beta^{2}-\alpha\gamma\right)X^{2}$ …
  14. Let …
  15. Let $S$ be the set containing all $3\times3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal …
  16. Let $A=\begin{bmatrix}\cos\theta&0&-\sin\theta\\0&1&0\\\sin\theta&0&\cos\theta\end{bmatrix}$. If for some $\theta\in(0,\pi)$, $A^{2}=A^{T}$, then the sum of the diagonal …
  17. Let $A=\begin{bmatrix}0&1&2\\a&0&3\\1&c&0\end{bmatrix}$, where $a,c\in\mathbb{R}$. If $A^{3}=A$ and the positive value of $a$ belongs to the interval $(n-1,n]$, where …
  18. If $A=\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix}$ and $M=A+A^2+A^3+\ldots+A^{20}$, then the sum of all the elements of the matrix $M$ is equal to ________.
  19. Let $A=I_2-2MM^{T}$, where $M$ is a real matrix of order $2\times1$ such that the relation $M^{T}M=I_1$ holds. If $\lambda$ is a real number such that the relation …
  20. The number of matrices $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d\in\{-1,0,1,2,3,\ldots,10\}$, such that $A=A^{-1}$, is ________.
  21. Let $A=\begin{bmatrix}1&-1\\2&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}\beta&1\\1&0\end{bmatrix}$, $\alpha,\beta\in\mathbf{R}$. Let $\alpha_1$ be the value of $\alpha$ …
  22. The number of all $3\times3$ matrices $A$, with enteries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $AA^T$ is $3$, is ________.
  23. Let $M=\left\{A=\begin{pmatrix}a&b\\c&d\end{pmatrix}:a,b,c,d\in\{\pm3,\pm2,\pm1,0\}\right\}$. Define $f:M\to\mathbf{Z}$, as $f(A)=\det(A)$, for all $A\in M$, where …
  24. Let $A=\begin{pmatrix}1+i&1\\-i&0\end{pmatrix}$ where $i=\sqrt{-1}$. Then, the number of elements in the set $\{n\in\{1,2,\ldots,100\}:A^{n}=A\}$ is $\_\_\_\_\_\_$ .
  25. Let $I$ be an identity matrix of order $2\times2$ and $P=\begin{bmatrix}2&-1\\5&-3\end{bmatrix}$. Then the value of $n\in\mathbf{N}$ for which $P^{n}=5I-8P$ is equal to …
  26. Let $A$ be a matrix of order $2\times2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of $A$ is a prime number $p$, $2<p<8$, then the …
  27. The number of elements in the set $\left\{A=\begin{pmatrix}a&b\\0&d\end{pmatrix}:a,b,d\in\{-1,0,1\}$ and $(I-A)^{3}=I-A^{3}\right\}$, where $I$ is $2\times2$ identity …
  28. Let $A=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}$. Then the number of $3\times3$ matrices $B$ with entries from the set $\{1,2,3,4,5\}$ and satisfying $AB=BA$ is …
  29. Let $X=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $A=\begin{bmatrix}-1&2&3\\0&1&6\\0&0&-1\end{bmatrix}$. For $k\in\mathbb{N}$, if $X'A^kX=33$, then $k$ is equal to …
  30. Let $A=\begin{bmatrix}x&1\\1&0\end{bmatrix}$, $x\in\mathbf{R}$ and $A^4=[a_{ij}]$. If $a_{11}=109$, then $a_{22}$ is equal to ________.
  31. Let $A=\begin{bmatrix}2&-1\\1&1\end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^{n}$, then $n$ is equal to $\_\_\_\_\_\_$ .
  32. Let $A=\begin{pmatrix}2&-2\\1&-1\end{pmatrix}$ and $B=\begin{pmatrix}-1&2\\-1&2\end{pmatrix}$. Then the number of elements in the set $\{(n,m):n,m\in\{1,2,\ldots,10\}$ …
  33. Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$. If $\omega=\frac{\sqrt3i-1}{2}$, then the number of elements in the set …
  34. Let $M=\begin{bmatrix}0&-\alpha\\\alpha&0\end{bmatrix}$, where $\alpha$ is a non-zero real number and $N=\sum_{k=1}^{49}M^{2k}$. If $\left(I-M^{2}\right)N=-2I$, then the …
  35. Let $M=(a_{ij})_{3\times3}$, where $a_{ij}=\begin{cases}-1, & i=j\\ i^{2}-j^{2}, & i j\end{cases}$ $(i,j=1,2,3)$. If $X=\begin{bmatrix}1\\1\\1\end{bmatrix}$, then value …
  36. The number of singular matrices of order $2$, whose elements are from the set $\{2,3,6,9\}$, is $\_\_\_\_\_\_$ .
  37. Let $A=\begin{bmatrix}1&2\\1&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}3&3\\\beta&2\end{bmatrix}$. If $A^{2}-4A+I=O$ and $B^{2}-5B-6I=O$, then among the two statements …
  38. Let $S=\left\{A=\begin{bmatrix}a&b\\c&d\end{bmatrix}:a,b,c,d\in\{0,1,2,3,4\}\text{ and }A^{2}-4A+3I=O\right\}$ be a set of $2\times2$ matrices. Then the number of …
  39. Let $A=\begin{pmatrix}1&-1&1\\2&1&-3\\1&1&1\end{pmatrix}$ and $10B=\begin{pmatrix}4&2&2\\-5&0&\alpha\\1&-2&3\end{pmatrix}$. If $B$ is the inverse of matrix $A$, then …
  40. If $A^{2}-A+I=0$, then the inverse of $A$ is :
  41. If $A=\begin{bmatrix}a&b\\b&a\end{bmatrix}$ and $A^{2}=\begin{bmatrix}\alpha&\beta\\\beta&\alpha\end{bmatrix}$, then
  42. If $A$ is a $3\times3$ non-singular matrix such that $AA'=A'A$ and $B=A^{-1}A'$, then $BB'$ equals :
  43. If $B$ is a $3\times3$ matrix such that $B^{2}=0$, then $\det\left[(I+B)^{50}-50B\right]$ is equal to :
  44. Let $A=\begin{pmatrix}0&0&-1\\0&-1&0\\-1&0&0\end{pmatrix}$. The only correct statement about the matrix $A$ is
  45. Let $A=\begin{bmatrix}1&1&2\\-2&0&1\\1&3&5\end{bmatrix}$. Then the sum of all elements of the matrix …
  46. Let $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}a&0\\0&b\end{pmatrix}$, $a,b\in N$. Then
  47. If $A=\begin{bmatrix}2&-3\\-4&1\end{bmatrix}$, then $\operatorname{adj}\left(3A^2+12A\right)$ is equal to :
  48. Let $A$ be a $2\times2$ matrix with real entries. Let $I$ be the $2\times2$ identity matrix. Denote by $\operatorname{tr}(A)$, the sum of diagonal entries of $A$. Assume …
  49. If $A=\begin{bmatrix}1&2&2\\2&1&-2\\a&2&b\end{bmatrix}$ is a matrix satisfying the equation $AA^{T}=9I$, where $I$ is $3\times3$ identity matrix, then the ordered pair …
  50. If $A=\begin{bmatrix}1&2&x\\3&-1&2\end{bmatrix}$ and $B=\begin{bmatrix}y\\x\\1\end{bmatrix}$ be such that $AB=\begin{bmatrix}6\\8\end{bmatrix}$, then :

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